The evolution of rotating very massive stars with LMC composition

1.8 MB·English

by  K. Köhler

Astronomy&Astrophysicsmanuscriptno.p9 (cid:13)cESO2015
January16,2015
The evolution of rotating very massive stars with LMC composition
⋆
K.Köhler1,N.Langer1,A.deKoter2,3,S.E.deMink2,4,5,P.A.Crowther6,C.J.Evans7,G.Gräfener8,H.Sana9,D.
Sanyal1,F.R.N.Schneider1,J.S.Vink8
1 Argelander-InstitutfürAstronomiederUniversitätBonn,AufdemHügel71,53121Bonn,Germany
2 AstronomicalInstitute“AntonPannekoek”,AmsterdamUniversity,SciencePark904,1098XH,Amsterdam,TheNetherlands
3 InstituutvoorSterrenkunde,KULeuven,Celestijnenlaan200D,3011Leuven,Belgium
5 4 ObservatoriesoftheCarnegieInstitutionforScience,813SantaBarbaraSt,Pasadena,CA91101,USA
1 5 CahillCenterforAstrophysics,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
0 6 DepartmentofPhysics&Astronomy,UniversityofSheffield,Sheffield,S37RH
2 7 UKAstronomyTechnologyCentre,RoyalObservatoryEdinburgh,BlackfordHill,Edinburgh,EH93HJ,UK
n 8 ArmaghObservatory,CollegeHill,ArmaghBT619DG,UK
a 9 SpaceTelescopeScienceInstitute,3700SanMartinDrive,Baltimore,MD21218,USA
J
Receiveddate/Accepteddate
5
1
ABSTRACT

]
R
Context.Withgrowingevidencefortheexistenceofverymassivestarsatsubsolarmetallicity,thereisanincreasedneedforcorre-
S spondingstellarevolutionmodels.
. Aims.WepresentadensemodelgridwithatailoredinputchemicalcompositionappropriatefortheLargeMagellanicCloud.
h
Methods.Weuseaone-dimensionalhydrodynamicstellarevolutioncode,whichaccountsforrotation,transportofangularmomen-
p
tumbymagneticfields,andstellarwindmasslosstocomputeourdetailedmodels.Wecalculatestellarevolutionmodelswithinitial
-
o massesfrom70to500M⊙andwithinitialsurfacerotationalvelocitiesfrom0to550km/s,coveringthecore-hydrogenburningphase
r ofevolution.
t Results.Wefindourrapidrotatorstobestronglyinfluencedbyrotationallyinducedmixingofhelium,withquasi-chemicallyhomo-
s
a geneousevolutionoccurringforthefastestrotatingmodels.Above160M⊙,homogeneousevolutionisalsoestablishedthroughmass
[ loss,producing pureheliumstarsatcorehydrogen exhaustionindependent oftheinitialrotationrate.Surfacenitrogenenrichment
isalsofoundforslower rotators,evenforstarsthat loseonlyasmallfractionoftheirinitialmass.Formodelsabove∼ 150M at
⊙
1 zeroage,andformodelsinthewholeconsideredmassrangelateron,wefindaconsiderableenvelopeinflationduetotheproxim-
v ityofthesemodelstotheirEddingtonlimit.Thisleadstoamaximumzero-agemainsequencesurfacetemperatureof∼ 56000K,
4 at ∼ 180M , and toan evolution of stars in the mass range 50M ...100M to the regime of luminous blue variables in the HR
⊙ ⊙ ⊙
9 [spellout]diagramwithhighinternalEddingtonfactors.Inflationalsoleadstodecreasingsurfacetemperaturesduringthechemically
7 homogeneousevolutionofstarsabove∼180M .
⊙
3 Conclusions.Thecoolsurfacetemperaturesduetotheenvelopeinflationinourmodelsleadtoanenhancedmassloss,whichprevents
0 starsatLMCmetallicityfromevolvingintopair-instabilitysupernovae.Thecorrespondingspin-downwillalsopreventverymassive
. LMCstarstoproducelong-durationgamma-raybursts,whichmight,however,originatefromlowermasses.
1
0 Keywords. stars:massive-stars:evolution-stars:rotation-stars:abundances-stars:earlytype
5
1
:
v1. Introduction massesofupto300M forseveralstarsintheLargeMagellanic
⊙
i Cloud(LMC)basedontheirluminosities.
X
Massive stars, with initial masses above ∼ 8M⊙, are power- We present stellar evolution models of very massive ro-
r
afulcosmicengines(Bresolinetal.2008).Theyproducecopious tating core-hydrogen burning stars, with initial masses up to
amountsof ionizing photons,strong stellar winds, energeticfi- 500M .ThesemodelswillbeusedforcomparisonwiththeVLT
⊙
nalexplosions,andmostoftheheavyelementsintheUniverse. FLAMESTarantulasurvey(Evansetal.2011),whichobserved
The most massive amongst them conduct a life very close to morethan800Oandearly-Bstarsinthe30Doradusregionlo-
their Eddingtonlimit, andmay thusbe proneto becomeunsta- catedintheLargeMagellanicCloud,tostudytheeffectsofro-
ble.Theyarethoughttobeabletoproducethemostspectacular tationalmixingand masslosson theevolutionof verymassive
stellar explosions, like hypernovae,pair-instability supernovae, stars.
andlong-durationgamma-raybursts(Langer2012).
Stellarevolutionmodelsforverymassivestarshavealready
Whilethevalueoftheuppermasslimitofstarsispresently beenpresented(seeTable1inMaeder&Meynet(2011)andref-
uncertain (Schneideretal. 2014), there is a great deal of evi- erences therein). In particular, we refer to the rotating stellar
denceofstarswithinitialmasseswellabove100M inthelocal models of Crowtheretal. (2010); Yusofetal. (2013), who cal-
⊙
Universe.Anumberofclosebinarystarshavebeenfoundwith culatedmodelswithinitialmassesupto500M .Itistheaimof
⊙
component initial masses above 100M (Schnurretal. 2008, thispapertopresentagridofstellarevolutionmodelsthathasa
⊙
2009;Sanaetal.2013b).Crowtheretal.(2010)proposedinitial densespacinginmassandrotationrates,suchthatitissuitable
Articlenumber,page1of21
A&Aproofs:manuscriptno.p9
forforthcomingpopulationsynthesiscalculations.Inthissense, (Langeretal. 1983; Langer 1991). In addition to convection,
our models are an extension of the LMC modelsof Brottetal. convectivecoreovershootingisincludedwithα = 0.335lo-
over
(2011a) to higher masses. In the mass range considered here, cal pressure scale heights, as calibrated in Brottetal. (2011a)
stellarwindmasslossandtheproximitytotheEddingtonlimit withtherotationalpropertiesofB-typestars(Hunteretal.2008;
playprominentroles. Vinketal.2010)fromtheVLT-FLAMESsurvey.While noob-
InSect.2webrieflydescribetheemployedstellarevolution servationalcalibration of the overshootingparameter exists for
code, including the input parameters for our calculations. Fol- starsoftheconsideredmassrange,wepointoutthattheroleof
lowing that, we present and discuss the stellar evolution mod- overshootingin our models is minor because of the large con-
els in Sect. 3, including the evolution of their mass and their vectivecoremassfractionsofverymassivestars.
surfaceabundances,withemphasisonchemicallyhomogeneous Rotationalmixing(Hegeretal.2000)isconsideredwiththe
evolution, and we compare our models with previous work in efficiencyparametersof f = 0.0228and f = 0.1 (Brottetal.
c µ
Sect. 4. Section5 containsourshortsummary.In AppendixA, 2011a). The most significant process causing rotationally in-
wepresentisochronesderivedfromourmodels,asfunctionsof ducedmixinginourmodelsistheEddingtonSweetcirculation.
ageandinitialstellarrotationrate. Furthermore, the transport of angular momentum by magnetic
fieldsduetotheSpruit-Taylordynamo(Spruit2002)isapplied,
which is assumed here not to lead to an additionaltransportof
2. Inputphysicsandassumptions chemical elements (Spruit 2006). Since the magnetic torques
leadtoa shallowangularvelocityprofileinourmodels,theef-
Brottetal. (2011a) computed three grids of stellar evolution
fectsoftheshearinstability,althoughincluded,arequitelimited
modelsfordifferentmetallicities(Galactic,SMC,LMC)forini-
duringthemainsequenceevolution.
tialmassesupto60M ,whichwerecomparedwiththeresultsof
⊙
the VLT-FLAMES Surveyof Massive Stars (Evansetal. 2005,
2006). In particular, the convective core overshooting parame- 2.3.Massloss
ter and the efficiency parameters for rotationally induced mix-
The evolution of very massive stars is intimately connected to
ing used by Brottetal. (2011a) were calibrated to reproduce
their mass-loss behaviour. Mass loss of very massive stars, for
the results of this survey (Hunteretal. 2008). The dense spac-
which few observationalconstraintsexist, is a very active field
ing in initial mass and rotational velocity used for the grid of
ofresearch.Here,weprovideabriefoverviewofthemassloss
Brottetal.(2011a)openedthedoorforstatisticaltestsofthethe-
prescriptionsadoptedinourcalculations.
oryofmassivestarevolution(Brottetal.(2011b);Schneideret
From the zero-age main sequence up to a surface helium
al.,inpreparation).Here,weextendtheLMCgridofBrottetal.
fractionY =0.4,weusethemass-losspredictionsbyVinketal.
uptoinitialmassesof500M , usingthesamenumericalcode, s
⊙ (2000, 2001) for O and B stars. These rates are valid for stars
physicalassumptions,andasimilarlydensegridspacing.
ofonemillionsolar luminositiesorless, i.e.starsbelow80M
To calculate rotating stellar evolution models, we use our ⊙
(Mokiemetal.2007),whicharenotveryclosetotheirEdding-
one-dimensional hydrodynamic binary stellar evolution code
ton limit (i.e. have Γ <∼ 0.3; see Eq. 3). Empirical tests of
(BEC) which is described in Hegeretal. (2000);Petrovicetal.
these predictionsdependcritically on the presence and proper-
(2005),andYoonetal.(2012).Itcontainsadetailedtreatmentof
tiesofsmallscalestructuresintheoutflows,knownasclumping.
rotation,angularmomentumtransportdueto internalmagnetic
Mokiemetal.(2007)showedonthebasisofHαandHeiiλ4686
fields,andstellarwindmassloss.Thecodesolvesallfivestellar
analysis that the predicted rates agree with observations if the
structureequationsthroughoutthestellarinterior,includingthe
materialisconcentratedinclumpsthathavea3–4timeshigher
stellarenvelopeuptoaRosselandopticaldepthofτ=2/3.Con-
densitythaninasmoothoutflow.Otheranalyses,whichinclude
vectionisconsideredthroughoutthestarusingthenon-adiabatic
modellingofultravioletresonancelines,deriveclumpingfactors
mixing-lengththeory.Ourcodeissuitedtotreatingstarscloseto
that may reach values of 10 (e.g. Bouret (2004); Bouretetal.
theEddingtonlimit,andtodescribetheeffectsofenvelopeinfla-
(2005,2012);Fullertonetal.(2006)).Inthislastcase,theVink
tionwhichoccurinthissituation(Ishiietal.1999;Petrovicetal.
etal.prescriptionadoptedheremayoverestimate M˙ byabouta
2006).
factorof2.An improvedhydrodynamicaltreatmentshowsthat
fornormalO stars the mass loss rates may be somewhatlower
2.1.Chemicalcomposition (Müller&Vink2008;Muijresetal.2012).
Predictionsforstarsinthe40–300M rangehavebeenpre-
⊙
The initial chemical composition for our models is chosen ac- sented by Vinketal. (2011). Objects at the upper mass end of
cording to corresponding observations of young massive stars thisrangemayhaveaveryhighEddingtonfactorΓ.Ithasbeen
and of Hii-regions in the LMC (Brottetal. 2011a). We thus foundthat,atsolarmetallicity,thewindstrengthsagreewiththe
adopt initial mass fractions for hydrogen,helium, and the sum standard Vink et al. recipe for objects that have an Eddington
of all metals of X = 0.7391, Y = 0.2562, and Z = 0.0047, factorforThomsonscatteringofΓe <∼ 0.7.ForhigherΓe values
respectively,with a non-solarmetalabundancepattern(see Ta- thesenewpredictionsshowanupturninthemass-lossrate,lead-
bles 1 and 2 in Brottetal.). We note that the applied opacities ingtoratesthatarehigherbyuptoaboutafactorof2compared
and mass loss rates (see below) are scaled with the LMC iron totheVinketal.valuesusedhere.Theauthorsassociatetheup-
abundance, not with the total metallicity, which is reduced by turnwith thestellar windbecomingopticallythick,i.e.leading
0.35dexwithrespecttothatoftheSun. tospectralmorphologywhichistypicalforWolf-Rayetstarsof
nitrogensubclass(WN)showinghydrogenemissionlines.Inter-
estingly,thispredictedupturnmayhave beenconfirmedobser-
2.2.Convectionandrotationalmixing
vationally(Bestenlehneretal.2014).Therelationbetweenmass
Convection with a mixing-length parameter of α = 1.5 loss and Eddington factor has also been explored for late-WN
MLT
(Böhm-Vitense 1958; Langer 1991) is applied as well as starsbyGräfener&Hamann(2008)andGräfeneretal.(2011).
semi-convection with an efficiency parameter of α = 1 TheyfindabehaviourthatissimilartotheresultsofVinketal.
SEM
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K.Köhleretal.:Evolutionofrotatingverymassivestars
(2011),thoughtheonsetofWolf-Rayettypeoutflowsoccursat
lowerΓ values.Gräfener&Hamannhoweverreportatempera-
e 500
turedependenceoftheirmass-lossratesthatissteeperthanthat
ofVinketal.(2011).
400
s]
Since the mass loss predictions for large Eddingtonfactors m/ 300
cannot yet be implemented unambiguously into stellar evolu- [kS
tioncalculations,we extrapolatetheVinketal.(2001)ratesfor M
A 200
starsabove106L⊙.WenotethatCrowtheretal.(2010)foundthe vZ
Vink et al. rates to agree within errorbars with those observed
100
in stars of up to ∼ 107L found in 30 Doradus. For objects
⊙
in the range 60...100M the objects are close to the model-
⊙ 0
independent mass-loss transition point between optically thin
50 100 150 200 250 300 350 400 450 500
andthickwinds.Inthisrangemass-lossrateshaverecentlybeen
calibrated with an uncertaintyof only ∼30%(Vink&Gräfener MZAMS [M⊙]
2012).
Fig.1.Initialequatorialrotationalvelocityversusinitialmass.Eachdot
inthisdiagramrepresentstheevolutionarysequenceinourmodelgrid
The Vink et al. mass-loss prescription shows a bi-stability
withthecorrespondinginitialparameters.Greydotscorrespondtomod-
jumpatabout25000K,leadingtoanincreaseofthemass-loss
elspresentedinBrottetal.(2011a)aswellaspreviouslyunpublished
rate by a factor of5 for stars ofspectraltype B1.5or later. We
modelscalculatedbyI.Brott,whileblackdotsrepresentthe110newly
includethisbi-stabilityjumpinourcalculations(cf.Brottetal. computedevolutionarysequences.
(2011a)). Additionally the Nieuwenhuijzen&deJager (1990)
empirical mass-loss rate is applied to cope with an in-
crease in mass loss when approaching the Humphreys- 3. Results
Davidson limit (HD limit). The transition from Vinketal.
3.1.EvolutionarytracksintheHR-diagram
(2000, 2001) to Nieuwenhuijzen&deJager (1990) occurs at
any effective temperature smaller than 22000 K where the AselectionofstellarevolutiontracksispresentedinFig.2.The
Nieuwenhuijzen&deJager (1990) mass-loss rate exceeds the luminosityofthestellarmodelsforagiveninitialmassandini-
Vinketal.(2000,2001)mass-lossrate. tial surface rotational velocity is shown as a function of their
effectivetemperature.Tracksareshownforninedifferentinitial
To account for Wolf-Rayet mass loss, the Hamannetal. massesfrom60M to500M ,withinitialsurfacerotationalve-
⊙ ⊙
(1995) mass-loss rate divided by a factor of 10 is applied for locitiesof0km/s,400km/s,and500km/s.
the surface helium fraction Y ≥ 0.7. Figure 1 in Yoonetal.
s
(2010)showsthatthiscorrespondswelltotheWolf-Rayetmass-
loss rate proposed by Nugis&Lamers (2000) for Wolf-Rayet
masses in the range 5M to 20M . For surface helium mass
⊙ ⊙
fractions from 0.4 ≤ Y ≤ 0.7 the mass-loss rate is linearly
s
interpolated between either the Vinketal. (2000, 2001) or the
Nieuwenhuijzen&deJager (1990) mass-loss rate and that of
Hamannetal.(1995).
Weapplyamass-lossenhancementforstarsnearcriticalro-
tationasinYoon&Langer(2005) whichconsidersareduction
of the critical rotational velocity for stars near their Eddington
limit.Itisstillunclearwhetherrapidrotationperseleadstoan
enhanced mass loss (Müller&Vink 2014), but it appears rea-
sonabletoconsiderthatthemass-lossrateincreasesclosetothe
Eddingtonlimit,whichisindeedreachedsoonerforrotatingob-
jects(Langer1997).
Fig. 3. Evolutionary tracks of stars withan initial mass of 80M , for
⊙
initialrotationalvelocitiesof0,50,100,150,200,250,300,350,400,
450,and500km/s.Trackswithinitialvelocitiesthatareevenmultiples
2.4.Modelgrid
of50km/saredrawninblue,oddmultiplesinred.
Figure 1 gives an overviewof our grid of evolution models by Uptoaluminosityoflog L/L ≃ 6.5,atabout190M ,the
⊙ ⊙
indicatingtheinitialmassesandinitialsurfacerotationalveloci- effectivetemperatureoftheZAMSincreaseswithincreasinglu-
tiesofallcomputedmodelsequences.Becauseoftheincreasein minosity.Forhigherinitialmassesthisbehaviourchanges.The
theEddingtonfactorwithmass,wedecreasethemaximumini- ZAMSmovestowardslowereffectivetemperaturewithincreas-
tialrotationalvelocityforhighermassesinordertoavoidstrong ingluminosityasaresultofstarshavingsignificantlyincreased
rotationallyinducedmasslossalreadyonthezero-agemainse- radiiand as a consequenceof their proximityto the Eddington
quence(Langer1998).Whereasmostsequencesarecomputedto limit.ThiseffectisdiscussedinmoredetailinSect.3.6.
core-hydrogenexhaustion,some of the most massive and most Figure2showsonlythetracksoftheslowestandfastestro-
rapidlyrotatingmodelswerestoppedshortlybeforetheirprox- tatorsinourgrid.However,wepointoutthatintheHR-diagram,
imitytotheEddingtonlimitcausednumericaldifficulties. theevolutionarytracksdonotchangemuchbelowaninitialro-
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7.2
500 M⊙
7
400 M⊙
6.8 300 M⊙
6.6
200 M⊙
)⊙
L
L/ 6.4
g( 150 M⊙
o
l
6.2
100 M⊙
6
80 M⊙
70 M⊙
5.8
HD limit
60 M⊙
ZAMS
5.6
60 50 40 30 20 10 0
T [kK]
eff
Fig.2.EvolutionarytracksofmassivestarsduringtheircorehydrogenburningevolutionintheHertzsprung-Russelldiagram.Foreachselected
initialmass(aslabelled),tracksareshownforthreedifferentinitialsurfacerotationalvelocities,3 =0,400,500km/s,inblack,blue,andred,
ZAMS
respectively.Thetimedifferencebetweentwosuccessivedotsoneachtrackis105yr.Thezero-agemainsequenceisdrawnasagreendashedline.
Theendofthetrackscorrespondstotheterminalagemainsequence.TheapproximatelocationoftheHumphreys-Davidsonlimitisindicatedby
theblackdashedline(Humphreys&Davidson1994).
tational velocity of ∼ 250km/s. This is demonstrated by the (2000)mass-lossrecipeandtheassumedmass-lossenhancement
example of 80M models in Fig. 3, for which only the tracks forstarsneartheirEddingtonlimit(Sect.2).
⊙
with initial rotational velocities of ∼ 300km/s and higher de-
viatesignificantlyfromthetrackofthemodelwithoutrotation. Figure2 shows that the evolutionary tracks of even our
As shown in Sect. 3.3, this is due to the absence of significant slowly rotating models avoid the upper-rightcorner of the HR
rotationally induced mixing of helium below the threshold ro- diagram.Thisisremarkable,sinceourverymassivestarmodels
tational velocity for quasi-chemically homogeneous evolution evolveveryclosetotheirEddingtonlimit,whichleadstoanin-
(cf.Brottetal.(2011a)).Wenotethatalthoughtheevolutionary flationoftheenvelope(cf.Sect.3.6).Wefindthatthehighmass-
tracks of the sequences computed with initial rotational veloc- loss rates at temperaturesbelow ∼ 30000Klead to significant
ities below ∼ 250km/sare almost identical, the corresponding helium enrichments for all stars above ∼ 60M (cf. Sect. 3.3)
⊙
modelsshowsignificantdifferencesconcerningtheevolutionof such that, as core hydrogen burning continues, they evolve to-
the surface abundancesof trace elements, e.g. boronand nitro- wardshotterratherthancoolersurfacetemperatures.
gen(cf.Sect.3.3).
Figure2 also contains the empirical upper luminos-
Themodelswithoutsignificantmixingofhelium,inthemass ity boundary of stars in the Milky Way, as derived by
range between 60M and 80M , expand during core hydrogen Humphreys&Davidson (1994). As we see, our slowly rotat-
⊙ ⊙
burning to surface temperatures as low as 5000K. This occurs ing models do penetrate the Humphreys-Davidson (HD) limit
partlybecauseoftherelativelylargeamountofconvectivecore and spend a significant amountof time at cooler temperatures.
overshootinginourmodels.Asecondreasonisthatwhenthese Whether this prediction is in contradiction with observations
models evolve through core hydrogen burning, they approach for the LMC is currently unclear. The stellar statistics near the
theEddingtonlimitbecausetheirL/Mratioisincreasing,andat HD limit for LMC stars from published work is not verygood
thesametimetheirenvelopeopacityisincreasingascoolersur- (Fitzpatrick&Garmany1990).AndevenintheMilkyWay,stars
facetemperaturesareachieved(cf.Sect.3.6).Forhigherinitial above the HD limit are observed, the most prominent exam-
masses,theredwardevolutionistruncatedatT ≃25000Kbe- ple being ηCarinae with a luminosity of log L/L = 6.7 and
eff ⊙
causeofthebi-stabilitymass-lossenhancementintheVinketal. T ≃30000K(Smith2013).
eff
Articlenumber,page4of21
K.Köhleretal.:Evolutionofrotatingverymassivestars
Inanycase,ourmodelspredictashortlived(∼ 105yr)yel- stability jump (Vinketal. 1999), according to the Vink et al.
low or red supergiant phase of stars in the mass range 60M prescription(Fig.4).Thecorrespondingmasslossissointense
⊙
to 80M during which the core is still burning hydrogen. Our that the models above 100M stop evolving redward at this
⊙ ⊙
models obtain a stellar wind mass-loss rate of the order of stage,sincetheirsurfacesbecomestronglyhelium-enriched.For
10−4M yr−1 duringthis stage. However,in this partof the HR ourmostmassivemodels,the500M sequences,themaximum
⊙ ⊙
diagram,mass-lossratesareveryuncertain.Wenotethatinpar- mass-lossrateatthisstageisabout2·10−4M yr−1.
⊙
ticular higher mass-loss rates (which may be due to the pulsa- Oncethe surfaceheliummassfractionhasincreasedowing
tionalinstabilityofthese models;Sanyaletal.,in prep.)would to mass loss to more than 40%, the Wolf-Rayet mass-loss rate
leadtoshorterlifetimesinthisevolutionarystage. isphasedin,reachingitsfullstrengthata surfaceheliummass
Thehighertheinitialmassofourmodel,thehigheristheir fraction of 70%. As a consequence, our 100−500M models
⊙
convectivecoremassfraction.Whileitexceeds80%inzero-age lose50-80%oftheirinitialmassbeforecorehydrogenexhaus-
main sequence stars above 100M⊙, it reaches 90% at 300M⊙, tion.DuringthephaseofWolf-Rayetwinds,themodelsevolve
and500M⊙starswithaconvectivecoremassfractionof95%are atdecreasingluminosityand allaccumulatein a narrowregion
almostfullyconvective.Thisputstheconvectivecoreboundary intheHR-diagram(Fig.2).
faroutintothestellarenvelope,wherethepressurescaleheight The three differentphases of mass loss can be clearly seen
israthersmall.Consequently,convectivecoreovershootingdoes in both panels of Fig. 4. Again, for our most extreme models,
nothavethesameimportanceasatsmallerstellarmassandplays the slowly rotating stars with an initial mass of ∼ 500M , the
⊙
onlyaminorroleinmostofthemodelspresentedhere. largestobtainedmass-loss rate is ∼ 5·10−4M yr−1, according
⊙
The effect of rotation on the evolution of massive stars is to our Wolf-Rayet mass-loss prescription. At this moment, the
discussedpreviously,forexampleinMaeder&Meynet(2000); starshavealuminosityoflog L/L ≃ 7.1.Assumingaterminal
⊙
Heger&Langer(2000);Brottetal.(2011a);Chieffi&Limongi wind speed of 1000km/s, we compute a wind-momentum-to-
(2013). In our models, there are two main changes in the stel- photon-momentum ratio of η ≃ 2 for this situation. With the
larevolutiontracksintheHRdiagram.First,theeffectivegrav- same numbers, we obtain a wind-kinetic-energy-to-luminosity
ity is reduced as a result of the centrifugalacceleration, which ratio of 0.003. The corresponding wind darkening is therefore
leads to a decrease in the effective temperature and luminos- negligible(Heger&Langer1996)andsignificantlysmallerthan
ity of a star compared to a non-rotating model. Second, above inGalacticWolf-Rayetstars(Lucy&Abbott1993).
a threshold rotational velocity for which the timescale of rota- The models that evolve quasi-homogeneouslyfirst increase
tional mixing becomes smaller than the nuclear timescale, the their luminosity more than the models described above. Con-
stars evolve quasi-chemically homogeneously (Yoon&Langer sequently, their mass-loss rate becomes larger than that of the
2005; Woosley&Heger 2006), and the corresponding models inhomogeneousmodels.Thehomogeneousmodelsdonotreach
evolve directly towards the helium main sequence in the HR- thebi-stabilitylimitasthe increasedheliumsurfaceabundance
diagram(Brottetal.2011a). due to rotationalmixing keeps their surface temperature above
Botheffectsareclearlyvisibleinourmodels.Forstellarevo- 25000K.However,theycanreachtheWolf-Rayetstageearlier,
lutiontrackswithinitialrotationalvelocitiesbelow250km/s,the and in the end lose similar amounts of mass as the inhomoge-
firsteffectmentionedisdominant.Itcanberecognizedmosteas- neous models. During the Wolf-Rayet stage, there are no clear
ilyfromtheZAMSpositionofourstellarmodels(Figs.2and3), characteristicsthatcandistinguishbetweenthetwotypesofevo-
wherethemodelsaredimmerandcoolerthefastertheyrotate. lution.
The most rapidly rotating stellar models undergo quasi-
chemically homogeneousevolution. As displayed in Fig. 2 for
3 = 400,500km/s, those with initial masses below ∼
ZAMS
125M showa strongincreasein luminosityandeffectivetem-
⊙
perature.Formoremassivemodels,however,homogeneousevo-
lution also leads to higher luminosities, but the surface tem-
perature is decreased. Towards core helium exhaustion, strong
Wolf-Rayetmasslossleadsthesestarstolowerluminositiesand
largersurfacetemperatures(cf.Sect.3.2).Wereturntothedis-
cussionofquasi-chemicallyhomogeneousevolutioninSect.3.4.
IsochronesintheHR-diagrambasedontracksdiscussedhereare
presentedinAppendixA.
3.2.Masslossandsurfacerotationalvelocity
The evolution of the models for the most massive stars is
stronglyaffectedbymassloss.Accordingtoourmass-losspre-
scription(Sect.2),ourmodelsundergomasslossofthreediffer-
entstrengths.Initially,themass-lossrateproposedbyVinketal. Fig.5.Massasafunctionofeffectivetemperatureforthemodelswith
(2000, 2001) is used. For our adopted composition, the mass- initialmassesof60M⊙,70M⊙,and80M⊙,andforinitialrotationalve-
lossrateisinitiallyoftheorderof3·10−6M yr−1forthe100M locities of 100, 400, and 500km/s, as indicated. The time difference
⊙ ⊙
models,whileforthe500M modelsitis∼ 6·10−5M yr−1.At betweentwodotsalongeachofthetracksis105yr.
⊙ ⊙
this rate, only 10-20% of the initial mass will be lost over the
lifetimeofthestars(Fig.4,rightpanel). Figure5comparesthemassevolutionofthehomogeneously
Themodelsthatdonotexperiencequasi-homogeneousevo- andinhomogeneouslyevolving60M to80M models.Itshows
⊙ ⊙
lution undergo an increase in their mass-loss rate at effective thatinthismassrange,thehomogeneouslyevolvingmodelslose
temperatures of ∼ 25000K, which is referred to as the bi- moremass,withmorethanhalfofthemassbeinglostduringthe
Articlenumber,page5of21
A&Aproofs:manuscriptno.p9
Fig.4.Massasafunctionofeffectivetemperature(leftpanel)andasafunctionoftime(rightpanel).Stellarmodelsthatevolvetowardseffective
temperaturesbelow27000Kareaffectedbythebi-stabilityjumpleadingtoanincreaseinthemass-lossrate.Thisisfollowedbyachangeinthe
slopeofthemassasafunctionoftime.Theincreaseinmass-lossrateforrapidlyrotatingmodelsisrelatedtothechangetotheHamannetal.
(1995)mass-lossrecipe.
Wolf-Rayetphase.Theinhomogeneouslyevolvingmodelslose forthehomogeneouslyevolvingonesbothleadtoastrongspin-
almostequalamountsduringtheirevolutionthroughthehotpart downofthe stars towardsthe endof their core-hydrogenburn-
of the HR-diagram as at cool surface temperatures, where the ingevolution.In fact, thisis onlyavoidedforstars initially be-
lattermasslossoccursonatimescaleofonly105yr. low ∼ 30M (cf. Fig. 3 of Vinketal. (2010)). Since the mag-
⊙
Asaconsequenceoftheinitiallyrathermoderatestellarwind neticcouplinginourmodelsensuresclose-to-rigidrotationdur-
massloss,mostofourmodelsevolveataconstantrotationalve- ingcorehydrogenburning,thisimpliesthatallourLMCmod-
locityformostofthecorehydrogenburning.Inthiscasetheloss els above ∼ 80M lose so much angular momentum that they
⊙
of angular momentum through the stellar wind (Langer 1998) cannot be considered to produce candidates for long-duration
andtheincreasedmomentumofinertiaduetoexpansioniscom- gamma-raybursts. In that respect, models that undergochemi-
pensated by the transport of angular momentum from the con- callyhomogeneousevolutionoflowermassarebettersuited(cf.
tractingcoretotheenvelope.AsshowninFig.6,thisholdseven Sect.3.4).
forthemostmassivestarsinourgrid.Onlythemodelswhichun- Effective temperatures and radii of observed massive stars
dergochemicallyhomogeneousevolution,asaresultoftheirin- maybeaffectedbyopticallythickstellarwinds,makingthestar
creasedluminosityandmass-lossrate,showamoderatedecline appearlargerandcooler.WeuseEq.(14)fromLanger(1989)
of their surface rotationalvelocity beforeentering the phase of
Wolf-Rayetwinds.
κ|M˙| 3
τ(R)= ln ∞ (1)
4πR(3 −3 ) 3
∞ 0 0
500
to estimate the optical depth of the stellar winds of our stel-
lar models. Here, R designates the radius of the stellar model
400
without taking the wind into account. This equation is derived
60 M⊙
s] fromaβ-velocitylawwithβ = 1.Inthiscase,weusetheelec-
m/ 300 500 M⊙ 300 M⊙ 100 M⊙ tronscatteringopacity(κ = σ(1+X),whereσistheThomson
[kot scatteringcrosssection),anexpansionvelocity30 = 20km/sat
vr 200 thesurfaceofthestellarmodel,andaterminalwindvelocityof
3 =2000km/s.
∞
100 Figure7showstheestimatedopticaldepthofstellarwindsas
a function of time for severalmodel sequences. The behaviour
of the optical depths seen in the figure is mostly related to the
0
0 0.5 1 1.5 2 2.5 3 3.5 4 change in the mass-loss rate. The optical depth increases for
t [Myr] higherinitialmassandhigherrotationrateasaresultofacorre-
spondingincreaseinthemass-lossrate.Whilethesenumbersare
Fig.6.Thesurfacerotationalvelocityisshownasafunctionoftimefor
onlyapproximations,itshowsthatthewindsofthemostmassive
stellarmodelsequenceswithfourdifferentinitialmasses(asindicated),
stars might already be optically thick (τ > 1) on the zero-age
andforthreeinitialsurfacerotationalvelocities.Therapiddecreasein
rotationrateseenfor3 ≃50,200km/siscausedbytheenhancedmass mainsequence,whichimpliesthatspectroscopically,thesestars
i
loss at thebi-stabilityjump, whereas thesteep decline of the rotation may already show Wolf-Rayet characteristics at this time (see
velocityforthefastrotators(3 ≃ 500km/s)isaconsequenceofWolf- also Gräfener&Hamann (2008); Crowtheretal. (2010)). Fur-
i
Rayettypemassloss. thermore, while the stars of 100M and below are expected to
⊙
showopticallythinwindsformostofthecorehydrogenburning
The strong mass loss for T < 25000K for the inhomo- evolution,theopticaldepthsofthe windsformodelsforwhich
eff
geneouslyevolvingmodels,andtheWolf-Rayettypemassloss aWolf-Rayettypewindhasbeenassumedmaybequitelarge.
Articlenumber,page6of21
K.Köhleretal.:Evolutionofrotatingverymassivestars
homogeneity,sincethesemodelsalsospindownquitedramati-
8
v : cally(Fig.6).
ZAMS
7 0 km/s
Figure9(leftpanel)demonstratestheroleofrotationalmix-
≈ 500 km/s
6 ing of helium in greater detail using the example of 80M⊙ se-
quences.Inthemodelsinitiallyrotatingwith0,50,100,150,and
5 200km/s, such mixing is essentially negligible, and only mass
R) 4 loss can increase the helium surface abundance close to core
τ( hydrogenexhaustion, which is why all of these models evolve
3 500 M⊙ 300 M⊙ 100 M⊙ alongthesametrackintheHR-diagram(Fig.3).Themodelsbe-
tween250and400km/sstartoutwithchemicallyhomogeneous
2
evolution(thoughtheonewith3 = 250km/sonlyforashort
1 τ(R)=1 60 M⊙ amountof time),and truncatethreot,hiomogeneousevolutionlater
intimeforhigherinitialrotation.Themodelswhichinitiallyro-
0
0 0.5 1 1.5 2 2.5 3 3.5 4 tatewith about450and500km/sundergochemicallyhomoge-
t [Myr] neousevolutionalltheway(thoughwiththehelpofmasslossin
theend),andtheirevolutionintheHRdiagramisalsopractically
Fig.7.StellarwindopticaldepthaccordingtoEq.(1),forsomeofour
identical(Fig.3).
model sequences, shown as a function of time. Weshow eight stellar
models withfour different initial masses (60, 100, 300, 500M ), and Theevolutionofthesurfacenitrogenabundanceinourmod-
⊙
twoinitialsurfacerotationalvelocities(0,500km/s)inblackandred, elsofrotatingandmasslosingstarsfollowsdifferentrules.Dur-
respectively.Thelineforunitwindopticaldepthisplottedtofacilitate ing a phase of chemically homogeneous evolution, nitrogen is
thecomparison. quicklymixedfromthecoretothesurfaceestablishingtheCNO-
equilibriumvalueinatmosphericlayers(Figs.8and9,leftpan-
els). Mass loss can also lead to nitrogen enhancements, even
3.3.Surfaceabundanceevolution withoutrotationalmixing(Fig.8).Figure9showsthatastrong
nitrogen surface enhancement can be obtained even in models
Derivingmassesoragesofstarsusingstellarevolutiontracksor with initial rotational velocities well below the threshold value
isochronesintheHRdiagramisnolongerstraightforwardwhen required for chemically homogeneous evolution. For example,
rotationalmixingisimportant.Agivenpairofeffectivetemper- the80M modelswith3 =150 and200km/senrichnitrogen
⊙ rot,i
ature and luminositycan be explainedby severalcombinations atthe stellar surface byfactorsof 4 and12by rotation,i.e. be-
ofmasses,rotationalvelocities,andages.Touniquelydetermine foremasslosskicksin.Thisorderofmagnitudeofnitrogenen-
theinitialparametersandtheageofastar,itisnecessarytoin- richmentisasexpected,sinceourmodelswerecalibratedtoin-
cludeadditionalobservables.Themassfractionsorabundances creasesthesurfacenitrogenabundancebyafactorofabout3for
ofelementsatthestellarsurfacearetracersoftherotationalmix- stars of 13–15M for an initial rotationalvelocity of 150km/s,
⊙
ing.Theycanbeusedtouniquelyidentifytheinitialparameters androtationalmixingisstrongerinmoremassivestars.
and the age of a star in a three (or more)dimensionalspace of
Thereasonisthatnitrogen,carbon,andoxygen,withamax-
observables(cf. Köhleretal. (2012)). We thereforediscuss the
imummassfractionoflessthanonepercent,arejusttraceele-
influence of time, initial rotationalvelocity and initial mass on
ments,suchthatevensignificantinternalgradientsdonotleadto
thesurfacecompositionofourmodels.
strongµ-barriers.Therefore,at the beginningof corehydrogen
The rotationalmixing in our modelsis sensitive to the gra- burning,beforesignificantamountsofhydrogenhavebeencon-
dient of the mean molecular weight µ. Essentially, any signifi- verted to helium, but after the CNO-cycle has already strongly
cant µ-barrier preventsrotationalmixing. Therefore, mixing of enhanced the nitrogen abundance in the convective core, rota-
heliumcanonlyoccurin ourmodelsaslongastheyarequasi- tionalmixingcanbringnitrogenoutofthecoreintotheradiative
chemically homogeneous. This, in turn, requires the timescale envelopeandlateronalsotothesurface.
forrotationalmixingtobeshorterthanthenucleartimescale,as
The surface enrichment of helium and nitrogen explained
thestarattemptstoestablishaµ-barrieronthenucleartimescale.
above allows a simple understanding of the occurrence of sur-
Inthefastestrotators,e.g.inthe100M⊙ and300M⊙ models faceenrichmentsintheHRdiagram.AsshowninFig.10,stars
initially rotating with ∼ 400 and 500km/s, the surface helium initially rotating with rotational velocities of 200km/s or less
abundancegoesalmostallthewaytoY =1towardscorehydro- thatareless luminousthanlog L/L ≃ 6.2are notexpectedto
s ⊙
genexhaustion(cf.Fig.8).However,thesamefigureshowsthat showanyheliumsurfaceenrichment(i.e.Y < 0.28)aslongas
s
this evolution is truncated for the rapidly rotating 60M mod- their surface temperature is higher than ∼ 20000K. The same
⊙
els, at Y = 0.68 and Y = 0.80 for initial rotational velocities is true for stars initially rotating slower than 300km/s below
s s
of ∼ 400 and 500km/s, respectively. These models spin down log L/L ≃ 5.6, and for those initially rotating slower than
⊙
during core hydrogen burning (Fig. 6) such that the timescale 400km/s below log L/L ≃ 4.8. Similar higherthresholdscan
⊙
forrotationalmixingeventuallybecomeslargerthanthenuclear bereadofffromFig.10forlargersurfaceheliummassfractions.
timescale. GiventhatRamírez-Agudeloetal.(2013)foundthat75%ofall
Figure 8 shows that the non-rotating300M sequence also Ostarsrotateslowerthan200km/sandthatamongstthe31O2
⊙
evolvestoY =1.Thereasonisthelargeconvectivecorefraction toO5starsnonewasfoundtorotatefasterthan∼ 300km/s,the
s
of this model, and its large mass-loss rate. Whereas the model heliumenrichmentinLMCstarsbelow∼100M⊙ isexpectedto
always keeps a small radiative envelope mass, the equivalent bequitesmallduringthemain-sequencephase.
amountof mass is lost on a short timescale such that core and Incontrast,asmorethanhalfofallOstarswerefoundtoro-
surface abundancesbecome almost equal (cf. Sect. 3.4; Eq.2). tatefasterthan100km/s,nitrogenenrichmentbyatleastafactor
For the rapidly rotating 100M and 300M models, mass loss of2isexpectedtobealmostubiquitousabovelog L/L ≃ 6.0,
⊙ ⊙ ⊙
ratherthanrotationalmixingmusttakeovertoenforcechemical andquitefrequentinmain-sequenceOstarsingeneral(Fig.11).
Articlenumber,page7of21
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Fig.8.Helium(leftpanel)andnitrogen(rightpanel)surfacemassfractionasafunctionoftimeformodelsof60,100,and300M androtation
⊙
ratesexplainedinthefigurekey.Whileforthefastrotatorstheenhancementsaremostlyduetorotationalmixing,fortheslowlyrotatingmodels
theincreaseinheliumandnitrogeninthe100M and300M massmodelsissolelyduetomassloss.
⊙ ⊙
Fig.9.Helium(leftpanel)andnitrogen(rightpanel)surfacemassfractionasafunctionoftimeformodelsof80 M withapproximateinitial
⊙
rotationalvelocitiesof0,50,100,150,200,250,300,350,400,450,and500km/s,duringtheircorehydrogenburningevolution.
Changes in the amount of neon, sodium, aluminum, and
magnesiumatthesurfacealsooccur,indicatingthatinverymas-
sivestarstheMgAl-andtheNeNa-cyclesareactive.Inourmod-
els,theamountofmagnesiumisreducedwhilealuminumispro-
duced.Themassfractionofneonincreasesinourmodels,while
sodium decreases. For more details, we refer to the electronic
datapublishedwiththispaper.
Lithium, beryllium, and boron are destroyed in the stel-
lar interior (McWilliam&Rauch 2004) at temperatures above
∼ 2.5 · 106K for lithium, ∼ 3.5 · 106K for beryllium, and
∼ 5.0 · 106K for boron. In our models, temperatures below
5·106Kareonlyfoundintheouterenvelope.Thereforelithium,
Fig. 12. Kippenhahn-diagram for our 200M model with v ≃
⊙ ZAMS
beryllium,andboroncanexistintheouterenvelope,whilethey 300km/s. The black solid line gives the stellar mass as a function
aredestroyedindeeperlayers.Fortheslowestrotatingmodels, of time. Blue shading indicates thermonuclear energy generation (see
thesurfaceabundancesofthesethreeelementsremainconstant colourbartotherightoftheplot).Greenhatchedpartsshowconvective
untillayerswhichwereexposedtohighertemperaturesearlierin regions,andconvectivecoreovershootingisindicatedinred.Threedif-
theevolutionareexposedtothesurfaceduetomassloss.When ferentregimescanbedistinguishedaccordingtotherateatwhichthe
thathappens,themassfractionsofberyllium,boron,andlithium massoftheconvectivecoredecreases(seetext).
arequicklyreduced.Forfastrotatingmodels,rotationalmixing
and mass loss lead to a gradualdecrease in lithium, beryllium,
andboronovertime.
Articlenumber,page8of21
K.Köhleretal.:Evolutionofrotatingverymassivestars
Y=0.35 Y=0.7 Y=0.50
Y=0.28
Y=0.35
Y=0.28
Y=0.35
Y=0.28
Fig.10.EvolutionarytracksintheHRdiagramofstellarmodelsinitiallyrotatingwithapproximately200km/s,andwithinitialmassesof12,15,
20,25,30,40,50,60,70,80,100,125,150,175,200,230,260,300,400,and500M .Overlaidarelines(inblue)ofconstantheliumsurface
⊙
massfractionforY=0.28,0.35,0.50,and0.70.LinesofconstantheliumsurfacemassfractionforY=0.28,0.35correspondingtomodelswith
approximateinitialrotationalvelocitiesof300and400km/sarealsoshown.
Fig.11.EvolutionarytracksintheHRdiagramofstellarmodelsinitiallyrotatingwithapproximately200km/s,andwithinitialmassesof6,7,
8,9,10,12,15,20,25,30,40,50,60,70,80,100,125,150,175,200,230,260,300,400,and500M .Overlaidarelines(inblue)ofconstant
⊙
nitrogensurfacemassfraction,correspondingtonitrogenenhancementfactorsof2,5,10,and20,asindicatedinblue.Linesofconstantnitrogen
enhancementfactorsof2and3correspondingtomodelswithapproximateinitialrotationalvelocitiesof150and100km/sarealsoshown.
3.4.Chemicallyhomogeneousevolution
AsmentionedinSect.3.1,wecandivideourmodelsintothree
classes.Thefirstone(whichwecallClassO)likelycorresponds
tomoststarsinobservedstellarsamplesandcontainsthemodels
thatevolvein the normalway, i.e. in which the rotationallyin-
Articlenumber,page9of21
A&Aproofs:manuscriptno.p9
Thus,regardlessofrotation,themodelisextremelychemically
homogeneousduringthisphase.
Wedefineafourthtypeofevolution(TypeM)by
M
env <0.1τ , (2)
M˙ H
where τ is the hydrogen burning timescale. Since this condi-
H
tion ensures that the surface-to-centralhelium abundance ratio
is above ∼ 0.9 — i.e. similar to or smaller than in the case of
rotationally induced quasi-chemically homogeneous evolution
— we can subdivide our models into further classes. For ex-
ample, the 200M sequence discussed here can be considered
⊙
of Class HOM, as it first undergoes chemically homogeneous
evolution due to rotational mixing, then ordinarily, and finally
chemicallyhomogeneousevolutionduetomassloss.
ItisinstructivetoconsiderFig.14tounderstandwhichevo-
lutionary classes are realized by our models. The non-rotating
Fig. 13. Central Y and surface Y helium abundance of our 200M
c s ⊙ sequences with initial masses of 300M and 500M achieve
model with v ≃ 300km/s (cf. Fig. 12), and the ratio Y/Y as a ⊙ ⊙
ZAMS s c chemicalhomogeneityatacentralheliummassfractionof0.85
functionoftime.
and 0.75,respectively,and thusbelongto Class OM. The non-
rotating 100M sequence does not reach homogeneity. Conse-
⊙
quently, although it achieves a quite high final surface helium
ducedmixingofheliumisnegligible.Thesecondonedescribes massfractionof∼0.65,itbelongstoClassO.Themostrapidly
the modelsthatundergoquasi-chemicallyhomogeneousevolu- rotatingmodelsdepictedinFig.14nevershowasignificantdis-
tion(ClassH),whichcorrespondtotheinitiallyfastestrotators. crepancybetweentheir centralandsurfaceheliumabundances,
Asthethirdclass,wehavethemodelsthatstartoutevolvingho- which implies that they evolve from H-type to M-type evolu-
mogeneously,butwhichspindownsuchthattherotationalmix- tion and belong to Class HM. The stars rotating initially with
ingofheliumstops(ClassHO),afterwhichtheyevolveredward ∼300km/sallundergoanH→Otransition,wherethetwomore
in the HR diagram as the ordinary models. These models re- massiveonesmovefurtheron toTypeM evolution.As starsin
tainamemoryoftheirpasthomogeneousevolution,inthatthey the Classes HOM and HM both start out and end their chemi-
will keep an enhanced helium surface abundance and a higher callyhomogeneousevolution,wewillnotdistinguishthemfur-
luminosity-to-mass ratio (Langer 1992) compared to ordinary theranddesignatethembothasClassHM.
stars throughoutthe rest of their core hydrogenburning evolu-
tion.
1
v = 0 km/s
At the highest masses considered here (M>150M ), our ZAMS
∼ ⊙ 0.9 v ≈ 300 km/s
ZAMS
models may also undergoquasi-chemically homogeneousevo- v ≈ 500 km/s
ZAMS
lution without rotationally induced mixing, which is due to a 0.8
combinationofanextremelyhighfractionoftheconvectivecore 0.7
tothetotalstellarmassanda veryhighWolf-Rayettypemass-
loss rate. We illustrate this by the example of our 200M se- Ys 0.6
⊙
quencewithvZAMS ≃300km/sinFig.12.Forsuchverymassive 0.5
stars,theconvectivecorecomprisesamajorfractionofthestel-
0.4
larmass.
0.3
We can divide the core hydrogen burning evolution of this
model into three parts according to the mass evolution of the 0.2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
convective core. During the first part, the mass of the convec-
Y
tive core decreasesas a functionof time, butmore slowly than c
the mass of the star, such that the mass of the radiative enve-
lopedecreasesandtheconvectivecoremassfractionincreases. Fig. 14. The mass fraction of helium in the stellar core Yc and at the
surfaceY aredepictedforstellarmodelsof100,300,and500M and
Duringthisphase,themodelundergoesquasi-chemicallyhomo- s ⊙
rotationratesexplainedinthefigurekey.Foragivenrotationrate,ini-
geneousevolution(cf.Fig.13).Att ≃ 1.2Myr,themodeltran-
tiallymoremassivestarsincreasetheirsurfaceheliumabundancemore
sitionstoordinaryevolution,fromwhichtimeontheconvective
quickly.Every105yrthemodelsarehighlightedbyfilledcircles.Chem-
coremassdecreasessomewhatfasterthanthetotalmass.During
ically homogeneous evolution is indicated by equal changes in both
this second part of its evolution, the helium surface abundance massfractions and corresponds toaslope unity. At Y ≥ 0.8, all cal-
c
increases,eventhoughatamuchlowerratethanthecentralhe- culated stellar evolution models show an evolution toward the lineof
liumabundance(Fig.13).Finally,inpartthree,theWolf-Rayet equalY andY,causedbymassloss.
c s
masslosskicksin,whichleadstoaverysmallmassofthenon-
convectivestellarenvelope(M ≃ 5M ).Theamountofmass Thus, we have five different classes of evolution, H, HM,
env ⊙
corresponding to the non-convectiveenvelope is lost (at a rate HO,OM,andO,whereallbutthelastinvolvequasi-chemically
of∼ 210−4M yr−1) onatimescaleof∼20000yrwhichcorre- homogeneous evolution. Class O can be identified in Fig. 15
⊙
spondstoonlyabout1%ofthecorehydrogenburningtime,τ . (rightpanel),whichshowstheclose-to-finalsurfaceheliummass
H
Consequently,thesurfaceheliummassfractionduringthisstage fractionforall ourmodelsequences.Lookingatthe slowly ro-
isroughlyequalto(within1%)thecentralheliummassfraction. tating models, this figure shows that mass loss starts affecting
Articlenumber,page10of21